Problem: Determine how many solutions exist for the system of equations. ${-4x-2y = 4}$ ${-6x-3y = 9}$
Convert both equations to slope-intercept form: ${-4x-2y = 4}$ $-4x{+4x} - 2y = 4{+4x}$ $-2y = 4+4x$ $y = -2-2x$ ${y = -2x-2}$ ${-6x-3y = 9}$ $-6x{+6x} - 3y = 9{+6x}$ $-3y = 9+6x$ $y = -3-2x$ ${y = -2x-3}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -2x-2}$ ${y = -2x-3}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.